# American Institute of Mathematical Sciences

July  2017, 13(3): 1189-1211. doi: 10.3934/jimo.2016068

## A multi-objective approach for weapon selection and planning problems in dynamic environments

 1 College of Information System and Management, National University of Defense Technology, Changsha 410073, Hunan, China 2 Business School, Hunan University, Changsha 410082, Hunan, China 3 State Key Laboratory of Complex System Simulation, Beijing Institute of System Engineering, Beijing, China 4 College of Information System and Management, National University of Defense Technology, Changsha 410073, Hunan, China

* Corresponding author

Received  July 2015 Published  October 2016

Fund Project: The authors are supported by National Natural Science Foundation of China under Grants 71501181, 71401167, 71201169 and 71371067

This paper addresses weapon selection and planning problems (WSPPs), which can be considered as an amalgamation of project portfolio and project scheduling problems. A multi-objective optimization model is proposed for WSPPs. The objectives include net present value (NPV) and effectiveness. To obtain the Pareto optimal set, a multi-objective evolutionary algorithm is presented for the problem. The basic procedure of NSGA-Ⅱ is employed. The problem-specific chromosome representation and decoding procedure, as well as genetic operators are redesigned for WSPPs. The dynamic nature of the planning environment is taken into account. Dynamic changes are modeled as the occurrences of countermeasures of specific weapon types. An adaptation process is proposed to tackle dynamic changes. Furthermore, we propose a flexibility measure to indicate a solution's ability to adapt in the presence of changes. The experimental results and analysis of a hypothetical case study are presented in this research.

Citation: Jian Xiong, Zhongbao Zhou, Ke Tian, Tianjun Liao, Jianmai Shi. A multi-objective approach for weapon selection and planning problems in dynamic environments. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1189-1211. doi: 10.3934/jimo.2016068
##### References:
 [1] H. Abbass, A. Bender, H.H. Dam, S. Baker, J. Whitacre and R. Sarker, Computational scenario-based capability planning, Proceeding of GECCO'08, (2008), 1437-1444. doi: 10.1145/1389095.1389378. [2] K.P. Anagnostopoulos and G. Mamanis, A portfolio optimization model with three objectives and discrete variables, Computers & Operations Research, 37 (2010), 1285-1297. doi: 10.1016/j.cor.2009.09.009. [3] K.P. Anagnostopoulos and G. Mamanis, The mean-variance cardinality constrained portfolio optimization problem: An experimental evaluation of five multiobjective evolutionary algorithms, Expert Systems with Applications, 38 (2011), 14208-14217. doi: 10.1016/j.eswa.2011.04.233. [4] S. Arnone, A. Loraschi and A. Tettamanzi, A genetic approach to portfolio selection, Neural Network World, 3 (1993), 597-604. [5] M. Barlow, A. Yang and H.A. Abbass, A temporal risk assessment framework for planning a future force structure, Proceeding of CISDA 2007, (2007), 100-107. doi: 10.1109/CISDA.2007.368141. [6] J. Branke, B. Scheckenbach, M. Stein, K. Deb and H. Schmeck, Portfolio optimization with an envelope-based multi-objective evolutionary algorithm, European Journal of Operational Research, 199 (2009), 684-693. doi: 10.1016/j.ejor.2008.01.054. [7] J. Branke and D.C. Mattfeld, Anticipation and flexibility in dynamic scheduling, International Journal of Production Research, 43 (2005), 3103-3129. doi: 10.1080/00207540500077140. [8] L.T. Bui, M. Barlow and H.A. Abbass, A multi-objective risk-based framework for mission capability planning, New Mathematics and Natural Computation, 5 (2009), 459-485. doi: 10.1142/S1793005709001428. [9] A.F. Carazoa, T. Gómez, J. Molina, A.G. Hernández-Díaz, F.M. Guerrero and R. Caballero, Solving a comprehensive model for multiobjective project portfolio selection, Computers & Operations Research, 37 (2010), 630-639. doi: 10.1016/j.cor.2009.06.012. [10] T.-J. Chang, N. Meade, J.E. Beasley and Y.M. Sharaiha, Heuristics for cardinality constrained portfolio optimisation, Computers & Operations Research, 27 (2000), 1271-1302. doi: 10.1016/S0305-0548(99)00074-X. [11] W.-N. Chen, J. Zhang, H.S.-H. Chung, R.-Z. Huang and O. Liu, Optimizing discounted cash flows in project scheduling\-an ant colony optimization approach, IEEE Transactions on Systems, Man, and Cybernetics-Part C: Applications and Reviews, 40 (2010), 64-77. [12] S.V. deVonder, E. Demeulemeester and W. Herroelen, A classification of predictive-reactive project scheduling procedures, Journal of Scheduling, 10 (2007), 195-207. doi: 10.1007/s10951-007-0011-2. [13] K. Deb, A. Pratap, S. Agarwal and T. Meyarivan, A fast and elitist multiobjective genetic algorithm: NSGA-Ⅱ, IEEE Transactions on Evolutionary Computation, 6 (2002), 182-197. [14] K.U.B.R. N. Deb and S. Karthik, Dynamic multi-objective optimization and decision-making using modified nsga-ii: A case study on hydro-thermal power scheduling, Lecture Notes on Computer Science, 4403 (2007), 803-817. doi: 10.1007/978-3-540-70928-2_60. [15] K. Doerner, W.J. Gutjahr, R.F. Hartl, C. Strauss and C. Stummer, Pareto ant colony optimization: A metaheuristic approach to multiobjective portfolio selection, Annals of Operations Research, 131 (2004), 79-99. doi: 10.1023/B:ANOR.0000039513.99038.c6. [16] P. Fox, A theory of cost-effectiveness for military systems analysis, Operations Research, 13 (1965), 191-201. doi: 10.1287/opre.13.2.191. [17] F. Ghasemzadeh, N. Archer and P. Iyogun, A zero-one model for project portfolio selection and scheduling, Journal of Operations Research Society, 50 (1999), 745-755. [18] B. Golany, M. Kress, M. Penn and U.G. Rothblum, Network optimization models for resource allocation in developing military countermeasures, Operations Research, 60 (2012), 48-63. doi: 10.1287/opre.1110.1002. [19] M.A. Greiner, J.W. Fowler, D.L. Shunk, W.M. Carlyle and R.T. McNutt, A hybrid approach using the analytic hierarchy process and integer programming to screen weapon systems projects, IEEE Transactions on Engineering Management, 50 (2003), 192-203. doi: 10.1109/TEM.2003.810827. [20] W.J. Gutjahr, S. Katzensteiner, P. Reiter, C. Stummer and M. Denk, Multi-objective decision analysis for competence-oriented project portfolio selection, European Journal of Operational Research, 205 (2010), 670-679. doi: 10.1016/j.ejor.2010.01.041. [21] M. Helbig and A.P. Engelbrecht, Analysing the performance of dynamic multi-objective optimisation algorithms, IEEE Congress on Evolutionary Computation, (2013), 1531-1539. doi: 10.1109/CEC.2013.6557744. [22] S. Hiromoto, Fundamental Capability Portfolio Management, PhD thesis, Pardee RAND Graduate School, 2013. [23] M.T. Jensen, Improving robustness and flexibility of tardiness and total flow time job shops using robustness measures, Applied Soft Computing, 1 (2001), 35-52. doi: 10.1016/S1568-4946(01)00005-9. [24] J. Kangaspunta, J. Liesiö and A. Salo, Cost-efficiency analysis of weapon system portfolios, European Journal of Operational Research, 223 (2012), 264-275. doi: 10.1016/j.ejor.2012.05.042. [25] T. Kremmel, J. Kubalik and S. Biffl, Software project portfolio optimization with advanced multiobjective evolutionary algorithms, Applied Soft Computing, 11 (2011), 1416-1426. doi: 10.1016/j.asoc.2010.04.013. [26] J. Liesiö, P. Mild and A. Salo, Robust portfolio modeling with incomplete cost information and project interdependencies, European Journal of Operational Research, 190 (2008), 679-695. doi: 10.1016/j.ejor.2007.06.049. [27] K. Metaxiotis and K. Liagkouras, Multiobjective evolutionary algorithms for portfolio management: A comprehensive literature review, Expert Systems with Applications, 39 (2012), 11685-11698. doi: 10.1016/j.eswa.2012.04.053. [28] D. Ouelhadj and S. Petrovic, A survey of dynamic scheduling in manufacturing systems, Journal of Scheduling, 12 (2009), 417-431. doi: 10.1007/s10951-008-0090-8. [29] A. Ponsich, A.L. Jaimes and C.A.C. Coello, A survey on multiobjective evolutionary algorithms for the solution of the portfolio optimization problem and other finance and economics applications, IEEE Transactions on Evolutionary Computation, 17 (2013), 321-344. doi: 10.1109/TEVC.2012.2196800. [30] K. Shafi, A. Bender and H.A. Abbass, Fleet estimation for defence logistics using a multi-objective learning classifier system, Proceeding of GECCO'11, (2011), 1195-1202. doi: 10.1145/2001576.2001738. [31] K. Shafi, A. Bender and H.A. Abbass, Multi objective learning classifier systems based hyperheuristics for modularised fleet mix problem, Proceeding of SEAL 2012, 7673 (2012), 381-390. doi: 10.1007/978-3-642-34859-4_38. [32] F. Streichert, H. Ulmer and A. Zell, Comparing discrete and continuous genotypes on the constrained portfolio selection problem, Genetic and Evolutionary Computation Conference, 3103 (2004), 1239-1250. doi: 10.1007/978-3-540-24855-2_131. [33] F. Streichert, H. Ulmer and A. Zell, Evaluating a hybrid encoding and three crossover operators on the constrained portfolio selection problem, IEEE Congress on Evolutionary Computation, 1 (2004), 932-939. doi: 10.1109/CEC.2004.1330961. [34] H. Sun and T. Ma, A packing-multiple-boxes model for r & d project selection and scheduling, Technovation, 25 (2005), 1355-1361. doi: 10.1016/j.technovation.2004.07.010. [35] J.M. Whitacre, H.A. Abbass, R. Sarker, A. Bender and S. Baker, Strategic positioning in tactical scenario planning, Proceeding of GECCO'08, (2008), 1081-1088. doi: 10.1145/1389095.1389293. [36] M. Workshop, Capabilities Based Planning: The Road Ahead, Technical report, Institute for Defense Analyses, Arlington, Verginia, 2004. [37] J. Xiong, J. Liu, Y. Chen and H.A. Abbass, A knowledge-based evolutionary multi-objective approach for stochastic extended resource investment project scheduling problems, IEEE Transactions on Evolutioanry Computation, 18 (2014), 742-763. [38] J. Xiong, K. wei Yang, J. Liu, Q. song Zhao and Y. wuChen, A two-stage preference-based evolutionary multi-objective approach for capability planning problems, Knowledge-Based Systems, 31 (2012), 128-139. doi: 10.1016/j.knosys.2012.02.003. [39] S.-C. Yang, T.-L. Lin, T.-J. Chang and K.-J. Chang, A semi-variance portfolio selection model for military investment assets, Expert Systems with Applications, 38 (2011), 2292-2301. doi: 10.1016/j.eswa.2010.08.017. [40] E. Zitzler and L. Thiele, Multiobjective evolutionary algorithms: A comparative case study and the strength pareto approach, IEEE Transactions on Evolutioanry Computation, 3 (1999), 257-271. doi: 10.1109/4235.797969.

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##### References:
 [1] H. Abbass, A. Bender, H.H. Dam, S. Baker, J. Whitacre and R. Sarker, Computational scenario-based capability planning, Proceeding of GECCO'08, (2008), 1437-1444. doi: 10.1145/1389095.1389378. [2] K.P. Anagnostopoulos and G. Mamanis, A portfolio optimization model with three objectives and discrete variables, Computers & Operations Research, 37 (2010), 1285-1297. doi: 10.1016/j.cor.2009.09.009. [3] K.P. Anagnostopoulos and G. Mamanis, The mean-variance cardinality constrained portfolio optimization problem: An experimental evaluation of five multiobjective evolutionary algorithms, Expert Systems with Applications, 38 (2011), 14208-14217. doi: 10.1016/j.eswa.2011.04.233. [4] S. Arnone, A. Loraschi and A. Tettamanzi, A genetic approach to portfolio selection, Neural Network World, 3 (1993), 597-604. [5] M. Barlow, A. Yang and H.A. Abbass, A temporal risk assessment framework for planning a future force structure, Proceeding of CISDA 2007, (2007), 100-107. doi: 10.1109/CISDA.2007.368141. [6] J. Branke, B. Scheckenbach, M. Stein, K. Deb and H. Schmeck, Portfolio optimization with an envelope-based multi-objective evolutionary algorithm, European Journal of Operational Research, 199 (2009), 684-693. doi: 10.1016/j.ejor.2008.01.054. [7] J. Branke and D.C. Mattfeld, Anticipation and flexibility in dynamic scheduling, International Journal of Production Research, 43 (2005), 3103-3129. doi: 10.1080/00207540500077140. [8] L.T. Bui, M. Barlow and H.A. Abbass, A multi-objective risk-based framework for mission capability planning, New Mathematics and Natural Computation, 5 (2009), 459-485. doi: 10.1142/S1793005709001428. [9] A.F. Carazoa, T. Gómez, J. Molina, A.G. Hernández-Díaz, F.M. Guerrero and R. Caballero, Solving a comprehensive model for multiobjective project portfolio selection, Computers & Operations Research, 37 (2010), 630-639. doi: 10.1016/j.cor.2009.06.012. [10] T.-J. Chang, N. Meade, J.E. Beasley and Y.M. Sharaiha, Heuristics for cardinality constrained portfolio optimisation, Computers & Operations Research, 27 (2000), 1271-1302. doi: 10.1016/S0305-0548(99)00074-X. [11] W.-N. Chen, J. Zhang, H.S.-H. Chung, R.-Z. Huang and O. Liu, Optimizing discounted cash flows in project scheduling\-an ant colony optimization approach, IEEE Transactions on Systems, Man, and Cybernetics-Part C: Applications and Reviews, 40 (2010), 64-77. [12] S.V. deVonder, E. Demeulemeester and W. Herroelen, A classification of predictive-reactive project scheduling procedures, Journal of Scheduling, 10 (2007), 195-207. doi: 10.1007/s10951-007-0011-2. [13] K. Deb, A. Pratap, S. Agarwal and T. Meyarivan, A fast and elitist multiobjective genetic algorithm: NSGA-Ⅱ, IEEE Transactions on Evolutionary Computation, 6 (2002), 182-197. [14] K.U.B.R. N. Deb and S. Karthik, Dynamic multi-objective optimization and decision-making using modified nsga-ii: A case study on hydro-thermal power scheduling, Lecture Notes on Computer Science, 4403 (2007), 803-817. doi: 10.1007/978-3-540-70928-2_60. [15] K. Doerner, W.J. Gutjahr, R.F. Hartl, C. Strauss and C. Stummer, Pareto ant colony optimization: A metaheuristic approach to multiobjective portfolio selection, Annals of Operations Research, 131 (2004), 79-99. doi: 10.1023/B:ANOR.0000039513.99038.c6. [16] P. Fox, A theory of cost-effectiveness for military systems analysis, Operations Research, 13 (1965), 191-201. doi: 10.1287/opre.13.2.191. [17] F. Ghasemzadeh, N. Archer and P. Iyogun, A zero-one model for project portfolio selection and scheduling, Journal of Operations Research Society, 50 (1999), 745-755. [18] B. Golany, M. Kress, M. Penn and U.G. Rothblum, Network optimization models for resource allocation in developing military countermeasures, Operations Research, 60 (2012), 48-63. doi: 10.1287/opre.1110.1002. [19] M.A. Greiner, J.W. Fowler, D.L. Shunk, W.M. Carlyle and R.T. McNutt, A hybrid approach using the analytic hierarchy process and integer programming to screen weapon systems projects, IEEE Transactions on Engineering Management, 50 (2003), 192-203. doi: 10.1109/TEM.2003.810827. [20] W.J. Gutjahr, S. Katzensteiner, P. Reiter, C. Stummer and M. Denk, Multi-objective decision analysis for competence-oriented project portfolio selection, European Journal of Operational Research, 205 (2010), 670-679. doi: 10.1016/j.ejor.2010.01.041. [21] M. Helbig and A.P. Engelbrecht, Analysing the performance of dynamic multi-objective optimisation algorithms, IEEE Congress on Evolutionary Computation, (2013), 1531-1539. doi: 10.1109/CEC.2013.6557744. [22] S. Hiromoto, Fundamental Capability Portfolio Management, PhD thesis, Pardee RAND Graduate School, 2013. [23] M.T. Jensen, Improving robustness and flexibility of tardiness and total flow time job shops using robustness measures, Applied Soft Computing, 1 (2001), 35-52. doi: 10.1016/S1568-4946(01)00005-9. [24] J. Kangaspunta, J. Liesiö and A. Salo, Cost-efficiency analysis of weapon system portfolios, European Journal of Operational Research, 223 (2012), 264-275. doi: 10.1016/j.ejor.2012.05.042. [25] T. Kremmel, J. Kubalik and S. Biffl, Software project portfolio optimization with advanced multiobjective evolutionary algorithms, Applied Soft Computing, 11 (2011), 1416-1426. doi: 10.1016/j.asoc.2010.04.013. [26] J. Liesiö, P. Mild and A. Salo, Robust portfolio modeling with incomplete cost information and project interdependencies, European Journal of Operational Research, 190 (2008), 679-695. doi: 10.1016/j.ejor.2007.06.049. [27] K. Metaxiotis and K. Liagkouras, Multiobjective evolutionary algorithms for portfolio management: A comprehensive literature review, Expert Systems with Applications, 39 (2012), 11685-11698. doi: 10.1016/j.eswa.2012.04.053. [28] D. Ouelhadj and S. Petrovic, A survey of dynamic scheduling in manufacturing systems, Journal of Scheduling, 12 (2009), 417-431. doi: 10.1007/s10951-008-0090-8. [29] A. Ponsich, A.L. Jaimes and C.A.C. Coello, A survey on multiobjective evolutionary algorithms for the solution of the portfolio optimization problem and other finance and economics applications, IEEE Transactions on Evolutionary Computation, 17 (2013), 321-344. doi: 10.1109/TEVC.2012.2196800. [30] K. Shafi, A. Bender and H.A. Abbass, Fleet estimation for defence logistics using a multi-objective learning classifier system, Proceeding of GECCO'11, (2011), 1195-1202. doi: 10.1145/2001576.2001738. [31] K. Shafi, A. Bender and H.A. Abbass, Multi objective learning classifier systems based hyperheuristics for modularised fleet mix problem, Proceeding of SEAL 2012, 7673 (2012), 381-390. doi: 10.1007/978-3-642-34859-4_38. [32] F. Streichert, H. Ulmer and A. Zell, Comparing discrete and continuous genotypes on the constrained portfolio selection problem, Genetic and Evolutionary Computation Conference, 3103 (2004), 1239-1250. doi: 10.1007/978-3-540-24855-2_131. [33] F. Streichert, H. Ulmer and A. Zell, Evaluating a hybrid encoding and three crossover operators on the constrained portfolio selection problem, IEEE Congress on Evolutionary Computation, 1 (2004), 932-939. doi: 10.1109/CEC.2004.1330961. [34] H. Sun and T. Ma, A packing-multiple-boxes model for r & d project selection and scheduling, Technovation, 25 (2005), 1355-1361. doi: 10.1016/j.technovation.2004.07.010. [35] J.M. Whitacre, H.A. Abbass, R. Sarker, A. Bender and S. Baker, Strategic positioning in tactical scenario planning, Proceeding of GECCO'08, (2008), 1081-1088. doi: 10.1145/1389095.1389293. [36] M. Workshop, Capabilities Based Planning: The Road Ahead, Technical report, Institute for Defense Analyses, Arlington, Verginia, 2004. [37] J. Xiong, J. Liu, Y. Chen and H.A. Abbass, A knowledge-based evolutionary multi-objective approach for stochastic extended resource investment project scheduling problems, IEEE Transactions on Evolutioanry Computation, 18 (2014), 742-763. [38] J. Xiong, K. wei Yang, J. Liu, Q. song Zhao and Y. wuChen, A two-stage preference-based evolutionary multi-objective approach for capability planning problems, Knowledge-Based Systems, 31 (2012), 128-139. doi: 10.1016/j.knosys.2012.02.003. [39] S.-C. Yang, T.-L. Lin, T.-J. Chang and K.-J. Chang, A semi-variance portfolio selection model for military investment assets, Expert Systems with Applications, 38 (2011), 2292-2301. doi: 10.1016/j.eswa.2010.08.017. [40] E. Zitzler and L. Thiele, Multiobjective evolutionary algorithms: A comparative case study and the strength pareto approach, IEEE Transactions on Evolutioanry Computation, 3 (1999), 257-271. doi: 10.1109/4235.797969.
Chromosome representation
A conceptual example of the adaptation process
A conceptual example of the recovery process
The whole adaptation procedure for each solution after dynamic changes occur
Non-dominated set obtained with mutation rate 0.5 and crossover rates varying from 0.6 to 1.0
Non-dominated sets obtained with crossover rate 0.7 and mutation rates varying from 0.1 to 0.6
Convergence graph using hypervolume measure over time in 30 runs
Comparison between non-dominated sets with and without the consideration of synergy effectiveness
Behavior of the non-dominated set in the presence of dynamic changes
Solutions in the presence of 1-4 dynamic changes without and after adaptation
Solutions in the presence of 5-8 dynamic changes without and after adaptation
A conceptual example of the calculation of adaptation effectiveness
Parameters of different type of weapons in the synthetical case
 $w$ $a^{low}_{w}$ $a^{up}_{w}$ $c_w$ $d_w$ $r_w$ 1 15 40 3 6 0 2 10 30 4 8 0 3 6 20 10 10 0 4 5 10 12 15 0 5 12 20 5 7 0 6 8 16 8 8 0 7 8 18 9 8 0 8 6 15 10 5 0 9 5 15 13 11 0 10 4 8 18 14 0 11 4 8 15 20 0 12 4 16 8 9 0 13 5 12 18 15 0 14 4 10 16 16 0 15 6 18 14 12 0 16 8 20 12 14 0 17 3 8 18 22 0 18 5 10 16 18 0 19 3 9 20 18 0 20 7 15 5 10 0
 $w$ $a^{low}_{w}$ $a^{up}_{w}$ $c_w$ $d_w$ $r_w$ 1 15 40 3 6 0 2 10 30 4 8 0 3 6 20 10 10 0 4 5 10 12 15 0 5 12 20 5 7 0 6 8 16 8 8 0 7 8 18 9 8 0 8 6 15 10 5 0 9 5 15 13 11 0 10 4 8 18 14 0 11 4 8 15 20 0 12 4 16 8 9 0 13 5 12 18 15 0 14 4 10 16 16 0 15 6 18 14 12 0 16 8 20 12 14 0 17 3 8 18 22 0 18 5 10 16 18 0 19 3 9 20 18 0 20 7 15 5 10 0
Parameters of dynamic environments
 $No.$ 1 2 3 4 5 6 7 8 $w$ 10 4 20 2 17 5 6 12 $t\_CW_w$ 22 26 28 30 35 48 50 54
 $No.$ 1 2 3 4 5 6 7 8 $w$ 10 4 20 2 17 5 6 12 $t\_CW_w$ 22 26 28 30 35 48 50 54
Correlation analysis between flexibility and adaptation in the presence of 8 changes
 $No.$ 1 2 3 4 5 6 7 8 $corrcoef$ 0.8559 0.7892 0.5409 0.3797 0.3233 0.8663 0.7933 0.4643 $P-value$ 0 0 0 0 0 0 0 0
 $No.$ 1 2 3 4 5 6 7 8 $corrcoef$ 0.8559 0.7892 0.5409 0.3797 0.3233 0.8663 0.7933 0.4643 $P-value$ 0 0 0 0 0 0 0 0

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